LAB 1-Measurements and Uncertainties Report

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Western Kentucky University *

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Chemistry

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Feb 20, 2024

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Measurement and Uncertainty Susie Mohamed Lab Partners: Lissette Perez and Christin Mikeal PHYS 232-501 01-31-23 ABSTRACT This lab was designed to understand the relationship between uncertainty and measuring instruments and to understand how error propagates. During this experiment, numerous measurements were taken of a 140-gram metal block, including the length, width, and thickness, with the uncertainty of each measurement mathematically calculated. These values were used to calculate the volume, which along with the previously mentioned mass of the block, were used to find a density value of 2.77 g/cm3. Using 2.7 g/cm3 as the accepted theoretical value density, a percent error of 2.59% was found for this experiment. Using the percent error, the experiment was found to be 97.41% accurate and using the calculated density, and its uncertainty, the experiment was found to be 99.997% precise. INTRODUCTION In order to calculate averages, uncertainties, volume, density, percent error, precision, accuracy, etc. for the experiement, it is necessary to use the equations below. Average ( x ¿ : ( Σ =sum of, N=number of measurements, x=measurement) Equation 1 x = ∑ i = 1 N xi N Uncertainty ( σ ): Equation 2 σ X = √ ∑ i = 1 N ( x − xi ) 2 N ( N − 1 ) Volume (V): (L=length, W=width, T=thickness) Equation 3

V=L x W x T Uncertainty in Volume: Equation 4 σ V =V √ ¿¿ Density: ( ρ =density, m=mass, V=volume) Equation 5 ρ = m V Uncertainty in Density: Equation 6 σ p =P √ ¿¿ Length is the average length plus or minus uncertainty, L= L ± σ L Equation 7 Width is the average width plus or minus uncertainty, W= W avg ± σ W Equation 8 Thickness is the average thickness plus or minus uncertainty, T= T avg ± σ T Equation 9 Volume is the average length times average width times average thickness, V=L L avg xW avg x T avg Equation 10

Precision of the density measurement, (1- σ p p ) x 100% Equation 11 Accuracy of the density measurement 100%-Percent error Equation 12 Initial length minus length average, L i − L avg ` Equation 13 Initial width minus width average W i − W avg Equation 14 Initial thickness minus average thickness T i − T avg Equation 15 MATERIALS AND METHODS Materials: Meter Stick, Vernier Caliper, metal block, scale, and Micrometer Caliper. Methods: Five trials each were performed for length, width, and thickness. For length, its trials consisted of using a meter stick to find the right and left edge in centimeters, the difference of the length between the two, the average of the five trials for the difference of length, deviation (initial length minus average length), deviation squared, and the square rooted deviation divided by number of trials. For width, its trials consisted of using a vernier caliper to find the width in centimeters, the average of the five trials of finding width, the deviation (initial width minus average width), deviation squared, and the square rooted deviation divided by number of trials. For thickness, its trials consisted of using a micrometer caliper to find the thickness in millimeters, the conversion of the measurements from millimeters to centimeters, the average of the five converted lengths, deviation (initial thickness minus average thickness), deviation squared, and the square rooted deviation divided by number of trials. After collecting the data,

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the mass of the block was measured on a scale and the data was inputted into equation four. The mass was later used to calculate density using the equation p=m/V. After, equation 6 was used to calculate the uncertainty of density. Precision, percent error, and density measurement were then calculated. RESULTS A) Length Table 1: Length Trial #1 Left Edge (cm) Right Edge (cm) Length L (cm) Deviation (cm) L i − L avg Deviation 2 ¿ ¿ 1 5.05 16.26 11.21 0 0 2 12.55 23.71 11.16 -0.05 0.0025 3 20.32 31.56 11.24 0.03 0.0009 4 27.25 38.45 11.20 -0.01 0.0001 5 40.12 51.38 11.26 0.05 0.0025 Table 1 displays five trials of length which consists of left edge, right edge, length, deviation, and the deviation squared. Deviation squared has units of centimeters squared (cm 2 ) and all other values have units of centimeters (cm). The left and right edges of the block were measured and with each trial, the left edge was subtracted from the right to give the length. Average length was found by adding all length values up and dividing by total number of values (5). Deviation was then found by subtracting the average length from the length of each trial and those values were then squared to find deviation squared. Using equation 2, uncertainty was found to be ±0.02 cm. ` Example Calculations: Finding Length (L) using Trial 1: 16.26 cm - 5.05 cm = 11.21 cm Finding Average Length ( L avg ):

11.21 cm + 11.16 cm + 11.24 cm + 11.20 cm + 11.26 cm = 11.21 cm Finding Deviation using Trial 2: L i − L avg 11.16 cm – 11.21 cm = -0.05 cm Finding Uncertainty ( σ ¿ : σ X = √ ∑ i = 1 N ( x − xi ) 2 N ( N − 1 ) σ X = √ ( 0 + 0.0025 + 0.0009 + 0.0001 + 0.0025 ) 20 σ X = ± 0.02 cm Length (L) Value: L = L ± σ L L=11.21 cm ± 0.02 cm B) Width Table 2: Width Trial #1 Width (cm) Deviation (cm) W i − W avg Deviation 2 ¿ ¿ 1 4.24 0.008 6.4 x 10 -5 2 4.25 0.002 4.0 x 10 -6 3 4.24 0.008 6.4 x 10 -5 4 4.25 0.002 4.0 x 10 -6 5 4.26 0.012- 1.44 x 10 -4

Table 2. charts five trials of width (width and deviation in centimeters (cm) and deviation squared with units of centimeters squared (cm 2 ) To find the average width, we added each trial which was 5 trials, then divided by 5. The width was measured five times with average width found by adding up all values and dividing by the number of trials (5). To find the deviation, we subtracted the average width from each of the 5 trials mentioned previously and those values were then squared to find deviation squared. Using equation 2, uncertainty was then found to be ±0.004 cm. Example Calculations: Finding Average Width ( W avg ¿ : 4.24 cm + 4.25 cm + 4.24 cm + 4.25 cm + 4.26 cm = 4.248 cm Finding Deviation using Trial 1: W i − W avg 4.24 cm – 4.248 cm =0.008 cm Finding uncertainty ( σ ¿ : σ X = √ ∑ i = 1 N ( x − xi ) 2 N ( N − 1 ) σ X = √ ( 6.4 × 10 − 6 ) + ( 4 × 10 − 5 ) + ( 6.4 × 10 − 6 ) +( 4 × 10 − 5 ) ( 1.44 × 10 − 4 ) 20 σ X = ± 0.004 cm Width (W) Value: W= W avg ± σ W W= 4.248 cm ± 0.004 cm

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C) Thickness Table 3: Thickness Trial No. i Thickness (mm) Thickness (1mm=0.1cm) Deviation (cm) T i − T avg Deviation 2 ¿ ¿ 1 10.66 1.066 0.006 3.6 x 10 -5 2 10.63 1.063 0.003 9.0 x 10 -6 3 10.66 1.066 0.006 3.6 x 10 -5 4 10.43 1.043 0.017 2.9 x 10 -4 5 10.63 1.063 0.003 9.0 x 10 -6 Table 3 displays five trials of thickness which consists of thickness in millimeters (mm), thickness in centimeters (cm), deviation (cm), and deviation squared with units of centimeters squared (cm 2 ). The thickness of the block was measured five times in mm then converted to cm using the conversion factor 1mm=0.1cm. The average thickness was found by adding up all values and dividing by each of the 5 trials. Deviation was found by subtracting the average thickness from the thickness value of each trial and those values were then squared to find deviation squared. Using equation 2, uncertainty was then found to be ±0.004 cm. Example Calculations: Finding average Thickness ( T avg ¿ : 1.066 cm +1.063 cm + 1.066 cm +1.043 cm + 1.063 cm =1.060 cm Finding Deviation using Trial 1: T i − T avg 1.066 cm – 1.060 cm =0.006 cm Finding uncertainty ( σ ¿ :

σ X = √ ∑ i = 1 N ( x − xi ) 2 N ( N − 1 ) σ X = √ ( 3.6 × 10 − 5 ) + ( 9.0 × 10 − 6 ) + ( 3.6 × 10 − 5 ) +( 2.9 × 10 − 4 ) ( 9.0 × 10 − 6 ) 20 σ X = ± 0.004 cm Thickness (T) Value: T= T avg ± σ T T = 1.066 cm ± 0.004 cm D) Volume Volume of the block was found using Equation 3: V=L x W x T =11.21cm x 4.248cm x 1.060cm V = 50.48 cm 3 Uncertainty of the volume was calculated using Equation 4: σ V =V √ ¿¿ σ V =V √ ¿¿ σ V = ±0.004 cm 3 Volume (V) Value: V= 50.48 cm 3 ± 0.004 cm 3 E) Mass Mass of the block was recorded as: M= 140 g ± 0.1 g

F) Density Density of the block was found using Equation 5: ρ = m V ρ = 140 g 50.48 cm 3 ρ =2.77 g cm 3 Uncertainty in the density was found using Equation 6: σ p = ρ √ ¿¿ σ p = ρ √ ¿¿ σ p = ± 0.00007 g cm 3 Density ( ρ ¿ Value: ρ = 2.77 g cm 3 ± 0.00007 g cm 3 Precision was calculated using Equation 11: P = ¿ ) x 100% P = ( 1 − 0.00007 2.77 ) x 100 P = 99.997% Percent Error was calculated using 2.7 g/cm 3 as the accepted theoretical value: % error = experimental − theoretial theoretical × 100% % error = 2.77 g / cm 3 − 2.7 g / cm 3 2.7 g / cm 3 × 100% % error = 2.59%

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Accuracy: Accuracy = 100% - Percent error =100% - 2.59% Accuracy = 97.41% DISCUSSION In this lab, we were able to use measuring equipment to be able to use equations to find measurements on an object (in this case, a metal block) to recognize how measurements, uncertainties, and errors are calculated. The results were length, width, and thickness allowed volume to be calculated which was then used to find the main data value, density, to be found. The results of the first three measurements including their uncertainties are as follows: L=11.21 cm ± 0.02 cm, W=4.248 cm ± 0.004 cm, and T = 1.066 cm ± 0.004 cm. The volume was 50.48 cm 3 ± 0.004 cm 3 and the density was 2.77 g/cm 3 ±0.00007 g/cm 3 . This value was very close to the theoretical value of 2.7 g/cm 3 and the uncertainty displayed more precise calculations. The experimental density value of 2.77 g cm 3 was very close to the theoretical value of 2.7 g cm 3 showing a high accuracy which was then confirmed with a mathematical equation. An accuracy of 97.41% and a calculated precision value of 99.93% using the uncertainty portion of the density value showed very little error throughout the calculations of the experiment. The percent error was a small value of 2.59%. Some possibly reasons to error in this experiment would be inaccurate measurements of length, width, and/or thickness based on the measuring equipment. For example, the micrometer caliper was dependent on how hard each lab partner turned the knob which would result in inconsistent answers for each trial. This would result in a slight offset of values which is a sequentially arrangement of equations that also lead to more inaccuracy. QUESTIONS 1. It is better to take multiple measurements in lab because it will not only help with accuracy, but it will help cut down on any possible errors/mistakes. 2. We calculate uncertainty in measurement because no matter how accurate you may think you are, humans make errors as does the equipment that is used to measure. The calculated uncertainty of a measurement helps the collected data become even more accurate than just the measurement itself.

3. The difference between a percent error and uncertainty of measurement is percent error is calculated using experimental data and theoretical data to determine the inaccuracy of a data as compared to the standardized data. Whereas the uncertainty of a measurement is used to express the range of accuracy for a measurement.